Apple cider vinegar works best to help the digestive system have a healthy load of good gut bacteria, IF and only IF it's organic. It also corrects the pH of the gut helping good bacteria to thrive. Good bacteria crowd out excessive amounts of bad bacteria. Additionally they nourish the bird by breaking down what the gizzard ground into food that can be absorbed, creating vitamins (especially those of the B variety), and making enzymes that help break down specific nutrients from food.
In other words, beneficial bacteria of the sort found in organic apple cider vinegar and probiotics (plain yogurt, acidophilus tablets/capsules, prepared livestock probiotics) both feed your bird and help protect them against illness.
However, as huge a fan as I am of ACV, I do not use it as a wormer. If you wanted to go natural, try other products as outlined in the other post.
In a case of an infestation serious enough to see worms or cause frothy diarrhea, the only recommend protocol is to actually worm the birds with a wormer. Then after worming start again with your more natural preventatives.
When you worm in the above case, you must use a gentle wormer first - THEN a stronger broad spectrum wormer. Starting with the strong one can kill a bird with dead-worm-overload or anaphylactic shock.
My recommended protocol for as natural a program as possible is this:
Worm the birds with wazine 17% (piperazine 17%, never the dog and cat kind) one day as directed on the bottle. Withhold eating all meat and eggs for 14 days. Then treat all birds with ivermectin drop on. Thereafter use natural preventatives (DE, cayenne, etc) and only worm twice a year with ivermectin. Ivermectin has a large safety margin and you don't have to withdraw from eating eggs/meat.